It is shown by Perrin-Riou and Schneider that if the \(p\) -adic regulator of \(E\)  is not zero, then the order of vanishing of \(f_{E}^{alg}(T)\)  is equal to the rank of \(E(\mathbb {Q})\)  and the leading term of \(f_{E}^{alg}(T)\)  is equal to \(p^{-g}\rho _p(E)\) , up to a \(p\) -adic unit. As a consequence, the \(p\) -adic Birch and Swinnerton-Dyer conjecture for \(E\)  follows from the main conjecture for \(E\) . This is where the truncated Euler characteristic intervenes. The truncated Euler-characteristic \(\chi _t(\Gamma , E)\)  is equal to the leading coefficient of \(f_E^{alg}(T)\)  up to a unit (see [1]}). As a consequence of the results of Schneider and Perrin-Riou mentioned above, \(\chi _t(\Gamma , E)=p^{-g}\rho _p(E)\) .
